A problem to simultaneously minimize values of two or more objective functions is known as a multiobjective optimization problem. Such a multiobjective optimization problem is used in a design stage of producing goods, and some conventional examples exist that the multiobjective optimization problem is applied to the design of a Static Random Access Memory (SRAM) and a slider in a Hard disk drive.
For example, in the design of SRAM, a problem to simultaneously minimize an objective function f1 (x1, x2) for the power and objective function f2 (x1, x2) for the size is solved in value ranges of variables x1 and x2.
There are a lot of techniques for solving such multiobjective optimization problems at high speed by numerical analysis.
On the other hand, there are some techniques for solving the multiobjective optimization problem by the computer algebra. In this method, computer simulation is carried out for various design parameter values to calculate output evaluation indicators for each case. Then, a model expression to approximate a relationship between the design parameters and output evaluation indicators is calculated to carry out optimization based on this model expression by using the computer algebra. As a processing for the optimization, there is a case where an expression representing a relationship between the cost and the performance is calculated from the obtained approximate expression and constraint conditions.
Incidentally, as for the computer algebra, a Quantifier Elimination (QE) method is known. This technique is a technique that an expression “∃x (x2+bx+c=0)”, for example, is changed to an equivalent expression “b2−4c≧0” by eliminating quantifiers such as “∃ and ∀”.
Specifically, the QE method is described in the following documents. However, because a lot of documents for the QE method exist, useful documents other than the following document exist. These documents are incorporated herein by reference.
Anai Hirokazu and Yokoyama Kazuhiro, “Introduction to Computational Real Algebraic Geometry”, Mathematics Seminar, Nippon-Hyoron-sha Co., Ltd., “Series No. 1”, Vol. 554, pp. 64-70, November, 2007, “Series No. 2”, Vol. 555, pp. 75-81, December, 2007, “Series No. 3”, Vol. 556, pp. 76-83, January, 2008, “Series No. 4”, Vol. 558, pp. 79-85, March, 2008, “Series No. 5”, Vol. 559, pp. 82-89, April, 2008.
Anai Hirokazu, Kaneko Junji, Yanami Hitoshi and Iwane Hidenao, “Design Technology Based on Symbolic Computation”, FUJITSU, Vol. 60, No. 5, pp. 514-521, September, 2009.
Jirstrand Mats, “Cylindrical Algebraic Decomposition—an Introduction”, Oct. 18, 1995.
It is possible to obtain solutions of the aforementioned multiobjective optimization problem by the aforementioned quantifier elimination method. However, the calculation efficiency of solving the aforementioned multiobjective optimization problem by the quantifier elimination method is bad, and the processing time becomes long.